Rank-one matrices

Recall that the rank of a matrix is the dimension of its range. A rank-one matrix is a matrix with rank equal to one. Such matrices are also called dyads.

We can express any rank-one matrix as an outer product.

Theorem: outer product representation of a rank-one matrix

Every rank-one matrix $A in mathbf{R}^{m times n}$ can be written as an ‘‘outer product’’, or dyad

A = p q^T,

where $p in mathbf{R}^m$ , $q in mathbf{R}^n$ .

Proof of the theorem.

The interpretation of the corresponding linear map x rightarrow y = Ax for a rank-one matrix is that the output is always in the direction , with coefficient of proportionality a linear function of : x rightarrow q^Tx .

We can always scale the vectors and in order to express as

A = sigma u v^T,

where $u in mathbf{R}^m$ , $v in mathbf{R}^n$ , with |u|_2 = |v|_2 = 1 , and sigma >0 .

The interpretation for the expression above is that the result of the map x rightarrow Ax for a rank-one matrix can be decomposed into three steps:

we project on the -axis, getting a number ;
we scale that number by the positive number ;
we lift the result (which is the scalar ) to get a vector proportional to .

See also: Single factor model of financial price data.